The Autonomous Intersection Control Method Based on Reduction in Vehicle Conflict Relationships

Abstract

Current autonomous intersection control strategies are facing issues, such as lack of foresight, frequent occurrence of deadlock, and low control system efficiency. To address these issues, a vehicle–road cooperative autonomous intersection control strategy based on reducing vehicle conflict relationships is proposed in this study. First, a conflict relationship graph that can describe the driving conflict relationship between vehicles is constructed. Second, the complement of the maximum clique in the conflict relationship graph is solved to determine the set of accepted vehicle reservation requests, enabling more vehicle reservation requests to be successfully processed in unit time while ensuring safe driving at the intersection and improving intersection throughput efficiency. Third, based on the maximum clique method, a taboo search method is used to search the neighborhood, thus improving the quality of the final solution with a smaller search cost. Simulation results show that compared to other control strategies, such as the FCFS (First Come First Served) strategy, the traffic signal control strategy (Traffic-Light), and the control strategy based on greedy algorithm search (Batch-Light), the proposed strategy can considerably reduce the average vehicle waiting time by 42%, 19%, and 10%, respectively, as well as increasing the number of vehicles passing through the intersection per unit of time by 35%, 20%, and 12%, respectively. These results demonstrate the effectiveness of the proposed strategy in improving the throughput of the intersection and reducing the average vehicle waiting time.

1. Introduction

Intersections are essential components of urban road networks, serving as important nodes and hubs that pose challenges to urban road traffic safety and congestion [1]. With the advent of connected and automated vehicles (CAVs) equipped with advanced autonomous driving and networking capabilities, they can substantially reduce intersection traffic accidents and reduce congestion [2]. Currently, research on intersection management systems for CAV vehicles can be broadly categorized into two categories: (1) CAV vehicles integrated with traditional traffic signal control systems, which utilize high-precision traffic data acquired by CAV vehicles to adjust signal control system phase and phase sequence schemes, catering to the needs of intelligent vehicles [3,4]; and (2) signal-free intersection management modes designed for fully automated driving vehicle scheduling, abandoning traditional signal control methods.

While the above research can enhance the efficiency of CAV vehicle passage, the signal control system is still primarily designed for human-driven vehicles, and its execution efficiency is not optimal for precisely controlled CAV vehicles. Therefore, some research institutions have shifted focus toward developing intersection control systems specifically designed for CAV vehicles. The Autonomous Intersection Management (AIM) autonomous intersection management system developed by Peter Stone’s team at the University of Texas at Austin [5] is one of the earliest systems that designed an FCFS control strategy with better performance and subsequently proposed various efficient control strategies [6], including the FCFS-Light control strategy supporting human-driven vehicles and the FCFS-Emerg strategy for emergency vehicle passage. Sascha et al. [7] designed an intersection control strategy based on combination auctions in the AIM system.

Subsequently, a crossroad control strategy that can effectively dispatch vehicles under dual constraints of state and control was designed based on centralized dispatching [8]. Using the Monte Carlo search method, a crossroad control strategy was established to accurately predict vehicle arrival times by predicting possible delays, thereby improving crossroad efficiency [9,10]. A crossroad control strategy was also developed in [11] using a greedy search algorithm to improve throughput by reducing conflicts between vehicles. Additionally, a multivehicle collaborative crossroad control strategy was designed based on linear programming [12,13]. A strategy based on mathematical planning is proposed in [14], which optimizes and coordinates the driving order of vehicles at intersections under physical and safety constraints to improve intersection efficiency.

A virtual platoon-based scheduling model was designed using a distributed control approach to achieve distributed optimization of intersection scheduling, as presented in [15]. By introducing explicit driving rules, a distributed multivehicle collaborative management system was established to improve the efficiency of intersection traffic by enforcing compliance with the rules [16]. In [17], the expected driving area occupied by shared vehicles at intersections and their expected arrival times were shared to coordinate the order of vehicle passage. A parameterized social force model is proposed in [18]. By adjusting the weights in the model, different sets of trajectory combinations can be obtained, and the optimal set of trajectory combinations can be selected to improve intersection capacity. A two-stage optimization and scheduling method for autonomous intersections is proposed in [19]. The first stage designs a time optimization model to give the arrival time of vehicles at intersections. The second stage proposes an optimization model for the trajectory of vehicle movement based on the output of the first stage to improve the overall intersection efficiency.

In [20], a study was conducted on intersection scheduling using deep learning. The authors proposed a framework based on convolutional neural networks to predict the average vehicle waiting time under different traffic densities. This prediction was used to select the least time-consuming strategy as the scheduling policy. In [21], an autonomous intersection control strategy based on reinforcement learning is proposed. This strategy formulates decisions by preserving both short-term and long-term objectives, allowing vehicles to pass through intersections in the shortest possible time.

The FCFS control strategy is currently an efficient strategy for intersection control, but it lacks consideration for changes in future traffic flow and is prone to deadlocks. The decision-making process also has a certain degree of blindness, which does not help maximize vehicle throughput at intersections. Traffic signal control strategies are designed for human-driven vehicles and are not efficient for CAVs, which can be precisely controlled. Intersection control strategies based on heuristic algorithms are prone to becoming stuck in local optima when traffic density is high and cannot further improve the efficiency of intersection traffic.

As the “brain” of the entire autonomous intersection management system, the control strategy directly affects the performance of the system. In order to address the issues with existing intersection control strategies, this paper proposes the CR (Conflict Reduction) autonomous intersection control strategy. Firstly, the spatiotemporal trajectories of vehicles are modeled. Secondly, a conflict relation graph is constructed to describe the driving conflict relations between vehicles. Thirdly, through three solving stages, the complement of the maximum clique in the conflict relation graph is solved as the set of accepted vehicle reservation requests, allowing more vehicle requests to successfully pass through the intersection in each batch processing cycle. Then, the taboo search method is used to search the neighborhood, and the quality of the final solution is improved with lower search costs while ensuring the safety of vehicles at the intersection and increasing the efficiency of intersection traffic. Finally, the effectiveness and reliability of the CR control strategy are demonstrated through simulation.

2. Question Description and Modeling

The autonomous intersection control system is a system designed to control intersections for CAVs that have full autonomous driving capabilities. Firstly, when the CAVs enter the intersection area, it begins sending requests to the roadside units to reserve passage through the intersection. Secondly, the autonomous intersection control system receives multiple requests, but due to the limited intersection space and time resources, the requests need to compete with each other, leading to conflicts. The system then applies the corresponding control strategy to determine the optimal passage sequence for CAVs in the intersection, resolving the conflicts and improving intersection throughput. Finally, after the CAVs leave the intersection area, communication with the roadside units is interrupted, as shown in Figure 1.

2.1. Batch Processing

The entire intersection control system is defined in the discrete time domain T, which is equally divided into processing periods of $T_i$. The system processes at time step t_j with a time step length of Δt_step. The time Axis T of the entire control system is an infinite set composed of batch processing periods $T_i$. Each batch processing period $T_i$ is a finite set composed of time steps $t_j$, as shown in Figure 2.

$$T=\left\{T_1,T_2,\dots ,T_i,T_{i+1},\dots T_k\right\}$$

(1)

$$T_i=\left\{t_j|j>0,t_{curr}\le t_j\le t_{last},t_{j+1}-t_j=\Delta t_{step}\right\}$$

(2)

where $t_{curr}$ represents the current time in the $T_i$ cycle, and $t_{last}$ represents the last time in the $T_i$ cycle.

In each batch processing cycle $T_i$, the autonomous intersection control system receives a set of reservation requests $R_{all}$ from vehicles that have made reservations to pass through the intersection.

From the set of vehicle requests to cross the intersection $R_{all}$, the set of requests $R_{re}$ that can pass through the intersection is selected.

$$\begin{gathered} R_{re}=\left\{v_i|1\le i\le n\le m,\hspace{1em}n,m\in N^{+}\right\}, \\ {R}_{re}\subseteq R_{all} \end{gathered}$$

(3)

2.2. Modeling Analysis

To reduce the computational complexity of the system, the spatial area of the rectangular intersection S is partitioned into a grid of $g\ast g$ two-dimensional subregions, with g typically set to 12. After partitioning, the i-th subregion of the intersection is denoted by $s_i$, such that the entire intersection $S$ can be regarded as a finite set composed of $s_i$, as shown in Figure 1.

$$S=\left\{s_i|1\le i\le g^2, g\in N^{+}\right\}$$

(4)

Let the estimated arrival time of the CAV vehicle at the intersection be denoted by $t_{arri}$ and the time when it leaves the intersection area be denoted by $t_{dep}$. At each time step $t_i$ during the vehicle’s travel within the intersection, the occupied grid combination is $s{s}_j$. The spatiotemporal network sequence $D_r$ describing the entire process of CAV occupancy in the intersection, can be defined as follows:

$$D_r=\left\{\left(t_i,s{s}_j\right)|t_i\in T,t_{arri}\le t_i\le t_{dep},s{s}_j\subseteq S\right\}$$

(5)

Therefore, the necessary and sufficient condition for a conflict relationship to occur between vehicles $v_i$ and $v_j$ that are scheduled to pass through an intersection can be defined as follows: during their passage through the intersection, $v_i$ and $v_j$ must occupy the same single or group of small grids at a certain moment. This means that the arrival time of both vehicles at the conflict point, denoted by $t_c$, must satisfy $t_c=t_i$ and $t_c=t_j$, and their respective driving domains, denoted by $D_r^i{\displaystyle \cap }{D}_r^j\ne \varnothing$, must have a nonempty intersection. Based on the identified conflict relationships between vehicles, a conflict matrix M can be constructed. The element in the conflict matrix M is set to 1 when there is a conflict between the trajectories of two vehicles and 0 when there is no conflict relationship.

$$M\left(i,j\right)=\{\begin{array}{c}1, \mathrm{the} \mathrm{conflict} \mathrm{relationship} \mathrm{between} v_i \mathrm{and} v_j, i\ne j\\ 0, \mathrm{others}\end{array}$$

(6)

2.3. Intersection Control Model

This paper aims to address the problem of n vehicles competing for intersection spatiotemporal resources and safely and efficiently passing through the intersection. To solve this problem, we adopt a batch processing approach and select the largest vehicle appointment request set, denoted by $\max (R_{re})$, from the set $R_{all}$ in each processing cycle. Subsequently, we establish the objective function f as follows:

$$\begin{gathered} f=\mathrm{Max}(R_{re})=\mathrm{Max}\left({{\displaystyle \sum }}_{i=1}^n{r}_i-{{\displaystyle \sum }}_{i=2}^n{r}_i{\displaystyle \cap }{r}_{i-1}\right) \\ s.t. n\le count\left(R_{con}\right) \\ {u}_i^t\ne 0, \forall i\in n \\ \frac{d{\theta }_i^t}{dt}\ne 0, \forall {\theta }_i^t\in \mathsf{\Theta } \end{gathered}$$

(7)

where ${{\displaystyle \sum }}_{i=1}^n{r}_i$ represents the set of vehicles chosen for processing from $R_{con}$ and ${{\displaystyle \sum }}_{i=2}^n{r}_i{\displaystyle \cap }{r}_{i-1}$ denotes the scenario in which the selected sets have conflicts with one another. $u_i^t$ signifies that a vehicle must not stop while passing through the intersection, and $d{\theta }_i^t/dt$ indicates that the driving direction of a vehicle cannot be changed. In other words, the entering and exiting directions of the intersection must be determined prior to entering it. $\mathsf{\Theta }$ represents the set of all possible directions.

3. Algorithm Implementation

This paper proposes a solution to the challenge of scheduling autonomous vehicles passing through intersections using the maximum clique method and taboo search algorithm. The solution process consists of two main steps: (1) determining the initial scheduling scheme based on the maximum clique method and (2) optimizing and adjusting the scheduling scheme using the taboo search algorithm.

3.1. Description of Autonomous Intersection Control Problem Based on Maximum Cliques

By transforming the conflict matrix, a conflict relation graph between vehicles can be established, consisting of nodes (CAVs) and arcs (conflict relations between vehicles), to form the conflict relation graph $G=\left(V,E\right)$. Here, V is the set of vehicle nodes in the reservation request set $R_{all}$, and E is the set of vehicle conflict relations, where there is an edge connecting vehicle nodes that have a conflicting reservation request.

In objective function f, the essence of seeking the maximum accepting set $\mathrm{Max}(R_{re})$ is to find the maximum number of nonconflicting vehicle pairs. Therefore, the conflict relation graph G can be transformed to obtain its complement graph $\overline{G}=\left(V,\overline{E}\right)$, where the node set V is still the set of vehicle nodes in the reservation request set $R_{all}$, and $\overline{E}$ can be understood as a set of compatible relationship edges connecting any two nonconflicting vehicle nodes. In the nonconflicting relation graph $\overline{G}$, a complete subgraph $sub\left(\overline{G}\right)=\left(V^{\ast },E^{\ast }\right)$ is defined, where $V^{\ast }$ is a subset of nodes in $\overline{G}$, and $E^{\ast }$ is a subset of edges in $\overline{E}$.$V^{\ast }=sub\left(V\right)$, $E^{\ast }=\overline{E}{\displaystyle \cap }sub\left(V\right)\times sub\left(V\right)$.

Based on the above analysis, this problem can be viewed as a multiresource combination optimization problem, and maximum clique theory is an effective way to describe and solve these types of problems [22]. The largest complete subgraph containing vehicle nodes in graph $\overline{G}$ is the maximum clique $cl{\left(\overline{G}\right)}_{max}$ that the control system needs to find, which satisfies the condition that the number of nodes in $\left|cl{\left(\overline{G}\right)}_{max}\right|$ is greater than $\left|cl(\overline{G}{)}_i\right|$. Therefore, the objective function f of the control system, based on some strategies, finds a maximum complete subgraph $cl{\left(\overline{G}\right)}_{max}$ that contains the largest number of vehicle nodes in the nonconflicting relation graph $\overline{G}$ to maximize the throughput of the intersection in each control cycle.

3.2. CR Policy

This study proposes a CR autonomous intersection control strategy for solving the maximum clique problem. The input of the CR control strategy is the nonconflicting relation graph $\overline{G}$ of vehicles, and the output is the maximum clique solved in the current cycle. The clique should include the maximum number of nonconflicting vehicles reserved in the current cycle. The process of finding the maximum clique in the CR control strategy is divided into three substages, and each stage aims to determine whether a clique of size tcs $\left(tcs\le \mathrm{count}(R_{all}\right))$, where $R_{all}$ is the set of all requests, is found. If found, the algorithm ends and outputs the set of tcs numbers. Otherwise, the search proceeds to the next stage. Finally, if no clique of size tcs is found, the currently found optimal result is output.

The three substages for finding the maximum compatible vehicle node set $\max (R_{re})$ are as follows:

(1)

Random subalgorithm stage: randomly select a vehicle reservation request node from the set, with no preference for selecting a vehicle node.

(2)

Degree subalgorithm stage: select a vehicle reservation request node with the maximum degree from the set (calculated initially).

(3)

Penalty subalgorithm stage: introduce a node penalty mechanism to diversify the search process and avoid search stagnation. Based on a greedy algorithm, points that are added more frequently to the current clique are less likely to be reselected in the future selection process.

The CR control strategy sets a maximum number of external loop iterations and selection attempts for each selection stage. The algorithm outputs the result when the maximum number of loop iterations or selection attempts is reached. Let $R_{re}^{temp}$ be the set of vehicles currently being considered by the system. To find the size of the current clique, let $A_0\left(R_{re}^{temp}\right)$ be the set of all nodes adjacent to all nodes in the current clique $R_{re}^{temp}$, and let $A_1\left(R_{re}^{temp}\right)$ be the set of all nodes connected to all nodes in $R_{re}^{temp}$ except for one node. Let U be the set of nodes that are added to $R_{re}^{temp}$ by adding one point from $A_1$($R_{re}^{temp}$) and deleting the point from $R_{re}^{temp}$ that does not satisfy the clique condition. U is initially empty.

For each external loop in every substage of the algorithm, an internal loop is set up. If the intersection of $A_0\left(R_{re}^{temp}\right)$ and U is not equal to $A_0\left(R_{re}^{temp}\right),$ then the intersection of the internal loop continues. The internal loop consists of two parts:

(1)

Inner subloop: a point is selected from $A_0\left(R_{re}^{temp}\right)$ and added to $R_{re}^{temp}$ according to the node selection strategy for the current substage. The internal loop count is updated, and if |$R_{re}^{temp}$| = $tcs$, the entire algorithm ends, and U is reset to empty.

(2)

Inner selection criteria: if the intersection of $A_1\left(R_{re}^{temp}\right)$ and U is not equal to $A_1\left(R_{re}^{temp}\right)$, the following process is executed: a point v is selected from $A_1\left(R_{re}^{temp}\right)$-($A_1\left(R_{re}^{temp}\right)$ and U) and added to $R_{re}^{temp}$. Points that are not adjacent to v are removed to satisfy the clique condition, and the set R_v of all points in $R_{re}^{temp}$ that are not adjacent to v is added to U. U is updated as U = U + $R_v$. The selection count is updated. After the internal loop body ends, the external loop count is incremented, and the penalty value is updated. A perturbation strategy is applied where a point is randomly selected and added to the current clique $R_{re}^{temp}$, and points that do not satisfy the condition are removed.

After the three substages of the CR control strategy are completed, the set of accepted vehicles $R_{re}$ is output, which represents the maximum set of accepted requests found during this cycle.

3.3. Solution Optimization Based on the Tabu Search Algorithm

Based on the optimal solution $R_{re}$ obtained from the maximum clique, a random accept vehicle reservation request is selected from the set to construct a neighborhood structure. Swapping it with a conflicting request can result in a feasible solution set. Among the candidate solution sets satisfying the nontabu or already-unlocked conditions in the neighborhood of the vehicle request set, the best solution is chosen as the current best solution and added to the tabu list. If the solutions in the candidate set are not better than the current solution, the best solution among them that is not tabu or already unlocked is chosen. If all neighborhood points are tabu, the best solution is pardoned, and the tabu list is updated. Through experimentation, it is set to terminate optimization when there is no improvement in the solution set 12 consecutive times.

3.4. Time Complexity Analysis

The best-case time complexity upper bound is $T_{best}=O\left(n\right)$;

The worst-case time complexity upper bound is $T_{worst}=O\left(n !\right)$;

The average-case time complexity upper bound is $T_{average}=O\left(n^2\right)$.

In the best-case scenario, when there are no conflicting nodes in the conflict graph, meaning that there are no conflicts between any two cars, the time complexity of this strategy is only $O\left(n\right)$.

In the worst-case scenario, during each search process, the algorithm will face O(n − 1) reduction choices. As a result, when the algorithm completes the final search, the number of combinations to solve will reach $O\left(n!\right)$.

In the general case, assuming the number of choices in each selection is $\overline{c}$, and considering that the cost of each search is $O\left(n\right)$, we can obtain the recursive formula for the time complexity of the algorithm in the general case as follows:

$$T\left(n\right)=O\left(n\right)+\overline{c}T\left(n-1\right)$$

(8)

Further derivation of the recursive formula above yields:

$$T\left(n\right)={\displaystyle \sum }_{i=1}^n{\overline{c}}^{n-i}O\left(i\right)$$

(9)

Considering that $\overline{c}$ is usually a constant close to $O\left(1\right)$ and that it is a very small value compared to the order of n, we can further transform the equation to:

$$T\left(n\right)={\displaystyle \sum }_{i=1}^{n}O\left(i\right)=O\left(n^2\right)$$

(10)

Therefore, the average-case time complexity upper bound of the algorithm is $T_{average}=O\left(n^2\right)$.

4. Experiments and Analysis

To validate the accuracy and execution efficiency of the CR autonomous intersection control strategy proposed in this study, the performance of the CR control strategy was evaluated using both offline and online simulation methods in the open-source autonomous intersection simulation platform AIM.

4.1. Performance Evaluation Analysis of the Offline Control Strategy

The evaluation indicators in the offline simulation included ① search time and ② efficiency of node search quantity. The intersection area was set as a square area, and the range of granularity g was from 12 to 24. Five hundred experiments were conducted for each granularity, and the execution efficiency of the CR control strategy under different granularity conditions was statistically analyzed. To better demonstrate the performance of the CR strategy, a global search control strategy (GS) that could search in a fully traversable manner was established and compared with the performance of the CR control strategy.

In Figure 3, the performance of the GS strategy is essentially equivalent to that of the CR control strategy when the order of the conflict matrix is less than 20. However, when the order of the conflict matrix exceeds 20, the CR control strategy outperforms the GS strategy. When the order of the conflict matrix reaches the experimental upper limit of 24, the CR control strategy requires only 0.00869 s of search time, which satisfies the real-time requirements of the control system. As shown in Figure 4, as the order of the conflict matrix increases, the search efficiency of the CR control strategy exhibits a downward trend, but the search rate still remains above 94.5%, which generally meets the performance requirements of the control system.

4.2. Analysis of Control Strategy Performance Based on Online Evaluation

(1) Simulation parameter settings

In the AIM system, a bidirectional six-lane square intersection area (g = 12) was defined with a lane width of 3.25 m. The number of vehicles generated followed a Poisson distribution, and the traffic flow distribution followed the Greenshields model. The vehicle speed was adjusted according to changes in traffic volume. The communication range of the intersection was set to a 300 m radius centered on the intersection’s midpoint. Two types of vehicles were modeled and generated according to a bimodal distribution (small car size: 4.3 m × 2.35 m, generation probability: 0.8; large car size: 10 m × 2.5 m, generation probability: 0.2). The time simulation step of the control system was set to 0.02 s, and the time slice was consistent with the simulation step. The system batch processing cycle was set to 2 s, and the experiment execution time was 1 h.

Under balanced and unbalanced traffic flow conditions, the performance of the CR control strategy was compared and evaluated against the FCFS, Batch-Light [11], and Traffic-Light control strategies.

(2) Indicators for online simulation performance evaluation

① Average Vehicle Waiting Time

This indicator represents the average time that a vehicle spends passing through the intersection in its current driving state.

$$\frac{{{\displaystyle \sum }}_{iϵn}{t}_i^s-{{\displaystyle \sum }}_{iϵn}{t}_i^c}{n}$$

(11)

where n is the total number of autonomous vehicles, $t_i^s$ is the actual time spent by the vehicle passing through the intersection under congested conditions, and $t_i^s$ is the time spent by the vehicle passing through the intersection in its current state under ideal conditions.

② Completed Vehicle Count at the Intersection

This indicator represents the total number of CAVs that safely pass through the intersection within a unit of time (1 h).

(3) Performance evaluation experiment of the control strategy for balancing vehicle flow

In the AIM simulation platform, the balancing vehicle flow situation represents that the flow of vehicles entering the intersection from the four directions is basically the same.

The arrival of vehicles at each entry direction of the intersection follows a Poisson distribution with a mathematical expectation of λ, where λ takes values of 0.1, 0.125, 0.15, and 0.175. The queueing situation of vehicles at the intersection was recorded for 2 min. As shown in Figure 5, the total number of vehicles queued under the CR control strategy is below 26 when λ takes values of 0.1, 0.125, and 0.15. When λ = 0.175, the number of queued vehicles shows an increasing trend, but it still remains below 80, demonstrating the strong robustness of the strategy.

At the beginning of each lane in every direction of the intersection, the vehicle flow was generated based on a Poisson distribution. The vehicle density ω changed from 0 veh·h⁻¹ (vehicles/hour) to 2500 veh·h⁻¹, and the vehicle’s forward direction was generated randomly based on traffic rules (e.g., for the rightmost lane, vehicles have two forward directions: straight or right turn). The traffic-light control strategy adopts a fixed phase timing scheme, as shown in

Table 1. Phase time in traditional traffic signal control.

Phase	Green (s)	Yellow (s)	Red (s)
WE	30	3	1
W	8	3	1
S	10	1	2
NS	40	3	2
N	10	1	2
E	8	3	1

.

As shown in Figure 6, when the traffic flow density is less than 240 veh·h⁻¹, the average waiting time of the CR control strategy and other strategies remains within an acceptable range. When the traffic flow density exceeds 300 veh·h⁻¹, the average waiting time of the FCFS control strategy increases substantially. When the vehicle density exceeds 2000 veh·h⁻¹, the average waiting time of the CR control strategy is lower than that of the other control strategies.

As shown in Figure 7, as the traffic flow density increases, the number of deadlocks in the FCFS control strategy also increases, and the number of completed vehicles increases relatively slowly. When the traffic flow density is less than 500 veh·h⁻¹, the performance of the CR control strategy is basically comparable to that of other strategies. When the vehicle density exceeds 1000 veh·h⁻¹, the CR control strategy is superior to the other strategies.

As shown in Figure 8, in terms of the number of vehicles served and rejected by the CR control strategy, as the traffic flow density increases, the number of vehicles served (i.e., completed) shows an increasing trend, but the rate of increase slows down when the vehicle density exceeds 1000 veh·h⁻¹. The number of rejected vehicles also increases with the vehicle density, and when the vehicle density is 2500 veh·h⁻¹, the number of rejected vehicles is 3656, thus showing a gradually increasing trend. This is because as the traffic flow density increases, the number of requests in each cycle increases, leading to a gradual decrease in the efficiency of finding the optimal solution, i.e., the maximum clique, in each cycle.

To validate the stability of the CR control strategy, we conducted 20 independent runs of the algorithm and averaged the results for comparison with the CR strategy without neighborhood search and the algorithm from reference [23], as shown in Table 1 and

Table 2. Simulation results at a traffic flow density of 1000 veh·h⁻¹.

Algorithm	Completed Vehicle	Convergence Iteration Count of an Algorithm
CR	10,011	19
CR (without Tabu)	9902	22
Reference [23]	9625	27

.

Through comparing the simulation results in Table 2 and

Table 3. Simulation results at a traffic flow density of 2000 veh·h⁻¹.

Algorithm	Completed Vehicle	Convergence Iteration Count of an Algorithm
CR	8105	17
CR (without Tabu)	7932	18
Reference [23]	7820	23

, it is evident that the strategy proposed in reference [23] exhibits the slowest convergence speed and yields inferior final solutions under different traffic flow conditions. On the other hand, the CR (without Tabu) control strategy is capable of finding more optimal solutions, although its convergence speed is slightly lower compared to the CR control strategy. Notably, the CR control strategy outperforms the others in terms of both convergence speed and the number of completed vehicles, making it the optimal choice.

(4) Performance evaluation experiment of control strategies under unbalanced traffic flow

At the beginning of each lane in the east–west direction of the intersection, traffic flow is generated based on the Poisson distribution, and the traffic volume ω varies from 0 veh·h⁻¹ to 2500 veh·h⁻¹, while the traffic volume density in the north–south directions is maintained at 1500 veh·h⁻¹. Other parameters are set to be the same as in the experiment with balanced traffic flow.

Figure 9 shows that under unbalanced traffic flow, the FCFS strategy is most affected by the unbalanced traffic flow in terms of the vehicle average waiting time indicator. When the vehicle density exceeds 500 veh·h⁻¹, the CR and tabu control strategies are generally comparable and better than the other control strategies, and when the vehicle density exceeds 1600 veh·h⁻¹, the average waiting time of the CR strategy is lower than that of the other strategies. Figure 10 compares the average waiting times of vehicles in the north–south and east–west directions for the CR and traffic-light control strategies. In the CR control strategy, the average waiting time of vehicles in the north–south direction is higher than that of vehicles in the east–west direction, but it is still better than the FCFS control strategy.

This study proposed an autonomous intersection control strategy based on the maximum clique theory. The strategy was evaluated through offline comparison with global search, assessment of vehicle queue lengths under different Poisson distribution parameter values, and comparison with other control strategies. The results demonstrate that the proposed strategy was effective and reliable in reducing vehicle delay and improving intersection throughput. Future work should focus on designing a strategy that can operate efficiently in wireless communication environments and ensure safety in real-world traffic conditions.

5. Conclusions

In this study, an autonomous intersection control strategy based on maximum clique theory was proposed. The strategy modeled vehicle spatiotemporal trajectories, established conflict relations between vehicles based on their trajectories, and abstracted the conflict relations into a graph. By solving the complement of the maximum clique in the conflict relation graph as the accepted vehicle reservation request set in each batch processing cycle, more vehicles could pass through the intersection in unit time, achieving the goal of reducing vehicle delay and improving intersection throughput.

Compared to the current research findings, our work mainly focuses on two aspects: (1) reducing the computational complexity of the autonomous intersection control system by gridifying the intersection area and dividing the control system timeline into equal batch processing cycles; and (2) using maximum clique theory to solve the autonomous intersection control problem, which has strong adaptive capabilities and can obtain optimal solutions in different traffic flow conditions, allowing more vehicles to safely pass through the intersection.

Achieving safe and efficient control of autonomous vehicles in an intelligent connected vehicle environment is a complex task. In addition to intelligent vehicles, another focus is internet-connected vehicles, which are also the key to achieving self-driving at the fifth level. Since this study was carried out in an ideal communication situation without considering the actual intersection communication environment, such as packet loss and delay, in the future, we will evaluate the impact of the wireless communication environment on the control system, designing an intersection control policy based on the wireless communication environment to improve the traffic efficiency of vehicles in the traffic environment approaching real intersections and ensure driving safety.